Notation for annihilator with ring and module interchanged?

Question

Let $M$ be an $R$-module. For a subset $S\subseteq R$, the set $$\{m\in M\mid (\forall s\in S)[sm=0]\}$$ is clearly a submodule of $M$. Is there a name or notation for it? It is essentially the definition of annihilator with ring and module interchanged.

Answer 1

The notations $(0:_M S)$ and $\operatorname{Ann}_M(S)$ are both commonly used for this submodule. I'm not sure there's a standard name, but I've heard it informally referred to as "the annihilator of $S$ in $M$" before. More generally, if $N$ is a submodule of $M$ and $S \subseteq R$ (of course, we can always replace $S$ by the ideal it generates), we write $(N:_M S):=\{m \in M \mid Sm \subseteq N\}$.